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IBPS Clerk Standard Deviation & Variance

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This page covers IBPS Clerk Standard Deviation & Variance with complete concept notes, 17 graded practice MCQs, key points and exam-specific tips. Free to study.

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Concept Notes

Standard Deviation & Variance— Rules & Concept

Core ConceptRead this first — the foundation of the topic

Standard Deviation and Variance are measures that tell us how spread out data is from the average. Think of them as measuring 'how different' the numbers are from each other. CORE CONCEPT:

Imagine you have marks of 5 students: 10, 20, 30, 40, 50. All are spread out widely. Now imagine: 28, 29, 30, 31, 32. These are clustered tightly around 30. Both have the same average (30), but the spread is different. Variance and Standard Deviation measure this spread. Variance (σ²) = Average of squared differences from the mean

Standard Deviation (σ) = Square root of Variance KEY RULES:

1. Standard Deviation is always non-negative (≥0) 2. If all numbers are identical, SD = 0

3. Larger SD means more scattered data; smaller SD means data is clustered 4. Standard Deviation is preferred over Variance because it's in the same units as original data

Formula BlockMemorise — at least one formula appears in every paper
Variance = Σ(x - mean)² / n
Standard Deviation = √Variance
For quick calculation: Variance = (Σx² / n) - (mean)²
Exam PatternsWhat examiners ask — read before attempting PYQs

SSC CGL typically asks: - Calculate SD or Variance from raw data - Compare spread of two datasets - Identify which dataset has more variation - Effect on SD when all values are multiplied or added by a constant SHORTCUT/TRICK: When values are multiplied by k: New SD = k × Original SD When values are added by c: New SD = Original SD (unchanged) This saves calculation time significantly.

Worked ExampleSolve this step-by-step before moving on
1
Step 1

Find mean = (2+4+6+8+10)/5 = 30/5 = 6

2
Step 2

Find differences from mean: (2-6)=-4, (4-6)=-2, (6-6)=0, (8-6)=2, (10-6)=4

3
Step 3

Square differences: 16, 4, 0, 4, 16

4
Step 4

Variance = (16+4+0+4+16)/5 = 40/5 = 8

5
Step 5

Standard Deviation = √8 = 2.83 (approximately)

Exam TrapsCommon mistakes students make — avoid these

Students forget to divide by n after summing squared differences. They calculate Σ(x-mean)² but stop there—this is NOT variance. You MUST divide by n.

Also, many confuse which measure to use; remember SD is more commonly reported in exams because it's interpretable.

Key Points to Remember

  • Variance measures average squared distance of data points from the mean; Standard Deviation is its square root.
  • Formula: Variance = Σ(x - mean)² / n; SD = √Variance
  • If all data values are identical, Standard Deviation = 0 (no spread).
  • When multiplying all values by k: New SD = k × Original SD; when adding c: SD remains unchanged.
  • Standard Deviation is preferred in exams because it's expressed in the same units as original data, making it more interpretable.
  • Larger SD indicates data is spread out; smaller SD indicates data is clustered around the mean.

Exam-Specific Tips

  • Standard Deviation formula: σ = √[Σ(x - μ)² / n], where μ is the arithmetic mean and n is number of observations.
  • Variance is the square of Standard Deviation: σ² = Variance.
  • Alternative variance formula for quick calculation: Variance = (Σx² / n) - (mean)².
  • Property: If each observation is multiplied by constant k, new SD = k × original SD (linear transformation rule).
  • Property: If constant c is added to each observation, SD remains unchanged (addition does not affect spread).
  • Standard Deviation of any dataset is always a non-negative value (σ ≥ 0).
  • Coefficient of Variation (CV) = (SD / Mean) × 100, used to compare variability across different datasets with different means.
  • For grouped data, class midpoints are used in calculations, and frequency weights are applied in variance formula.
Practice MCQs

Standard Deviation & Variance — Practice Questions

17graded MCQs · easy to hard · full solution & trap analysis

All MCQs →
Practice 1easy

The mean of five numbers is 20 and the sum of squared deviations from the mean is 500. What is the standard deviation?

Practice 2easy

The variance of a dataset is 64. If each value is decreased by 10, what is the new variance?

Practice 3easy

For a dataset, if each value is multiplied by 3, how does the standard deviation change?

Practice 4easy

The variance of a dataset is 36. What is the standard deviation?

Practice 5easy

A dataset has values: 2, 4, 6, 8, 10. What is the variance? (Mean = 6)

Practice 6easy

If the standard deviation of a set of numbers is 5, what is the variance?

Practice 7medium

The standard deviation of a dataset is 8. If 10 is subtracted from each observation, what is the new standard deviation?

Practice 8medium

The coefficient of variation of a dataset is 25%. If the mean is 80, what is the standard deviation?

Practice 9medium

The variance of a dataset is 36. If each observation in the dataset is multiplied by 5, what will be the new variance?

Practice 10medium

The standard deviation of five numbers is 4. If the sum of the numbers is 50, what is the sum of their squares?

Practice 11medium

Two datasets A and B have the same mean of 20. Dataset A has 8 observations with variance 9, and Dataset B has 12 observations with variance 4. What is the combined variance of both datasets?

Practice 12medium

A dataset has 6 observations: 4, 6, 8, 10, 12, 14. Calculate the variance of this dataset.

Practice 13hard

In a dataset of 20 observations, the sum of squared deviations from the mean is 1200. A new observation with value equal to the mean is added. What is the new variance?

Practice 14hard

A dataset contains 5 observations: 12, 18, 24, 30, 36. If each observation is multiplied by 3, what is the ratio of the new standard deviation to the original standard deviation?

Practice 15hard

A sample of 10 students has a mean score of 60 and variance of 36. If one student with score 48 is removed, what is the new variance? (Assume the removed student's score is exactly 1 SD below the mean.)

Practice 16hard

A frequency distribution has mean 40 and standard deviation 8. If all values are shifted by adding 15 to each observation, and then multiplied by 2, what is the new standard deviation?

Practice 17hard

Two datasets A and B have the same mean of 50. Dataset A has variance 64 and Dataset B has variance 100. If we combine both datasets (equal number of observations from each), what is the variance of the combined dataset?

60-Second Revision — Standard Deviation & Variance

  • Formula: Variance = Σ(x-mean)²/n; SD = √Variance. Use quick formula: Variance = (Σx²/n) - (mean)² to save time.
  • Multiply by k → SD multiplies by k; Add constant c → SD unchanged. Use this trick for transformation questions.
  • SD=0 only when all values are identical. Higher SD = more scattered; Lower SD = clustered around mean.
  • Trap: Don't forget to divide by n after summing squared differences—this is the most common error in calculations.
  • Remember: SD is preferred over Variance in SSC exams because it's in original units and easier to interpret.
  • For two datasets with same mean: compare SDs to determine which is more variable/consistent.
  • In exam, if asked 'which dataset varies more?'—calculate or compare SD values; higher SD = more variation.
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